Angle

In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices.

In some contexts, such as identifying a point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent.

A turn is a unit of plane angle measurement equal to 2degrees or 400 gradians.

The ratio of the length s of the arc by the radius r of the circle is the measure of the angle in radians.

The radian is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics.

The measure of the angle in another angular unit is then obtained by multiplying its measure in radians by the scaling factor k⁄2, where k is the measure of a complete turn in the chosen unit (for example 360 for degrees or 400 for gradians):

A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle, defined so that a full rotation is 360 degrees.

;Quadrant (n = 4): The quadrant is 1⁄4 of a turn, i.e. a right angle.

In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn.

Angle is also used to designate the measure of an angle or of a rotation.

In two dimensions, only a single angle is needed to specify a rotation about the origin – the angle of rotation that specifies an element of the circle group (also known as

Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles.

Angles are also formed by the intersection of two planes in Euclidean and other spaces.

. In particular, a reflex angle

An exterior angle of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle.

. . ) are also used, as are upper case Roman letters in the context of polygons.

In mathematics, a function f is cofunction of a function g if f(A) = g(B) whenever A and B are complementary angles.

The most often considered types of bisectors are the segment bisector (a line that passes through the midpoint of a given segment) and the angle bisector (a line that passes through the apex of an angle, that divides it into two equal angles).

Angle is also used to designate the measure of an angle or of a rotation.

Two edges meeting at a corner are usually required to form an angle that is not straight (180°); otherwise, the collinear line segments will be considered parts of a single side.

In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint.

radians, 180°, or 1⁄2 turn; the measures of the interior angles of a simple convex quadrilateral add up to 2

In a crossed quadrilateral, the four "interior" angles on either side of the crossing (two acute and two reflex, all on the left or all on the right as the figure is traced out) add up to 720°.

A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive.

A transversal is a line that intersects a pair of (often parallel) lines and is associated with alternate interior angles, corresponding angles, interior angles, and exterior angles.

By Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal.

;Sextant (n = 6): The sextant (angle of the equilateral triangle) is 1⁄6 of a turn.

In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°.

Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane.

where θ is the angle between A and B.

Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.

In spherical trigonometry, angles are defined between great circles.

is cyclic if and only if its opposite angles are supplementary, that is :